1 research outputs found
Approximate Frank-Wolfe Algorithms over Graph-structured Support Sets
In this paper, we propose approximate Frank-Wolfe (FW) algorithms to solve
convex optimization problems over graph-structured support sets where the
\textit{linear minimization oracle} (LMO) cannot be efficiently obtained in
general. We first demonstrate that two popular approximation assumptions
(\textit{additive} and \textit{multiplicative gap errors)}, are not valid for
our problem, in that no cheap gap-approximate LMO oracle exists in general.
Instead, a new \textit{approximate dual maximization oracle} (DMO) is proposed,
which approximates the inner product rather than the gap. When the objective is
-smooth, we prove that the standard FW method using a -approximate
DMO converges as in general, and as over a
-relaxation of the constraint set. Additionally, when the objective is
-strongly convex and the solution is unique, a variant of FW converges to
with the same per-iteration
complexity. Our empirical results suggest that even these improved bounds are
pessimistic, with significant improvement in recovering real-world images with
graph-structured sparsity.Comment: 30 pages, 8 figure